Understanding the uniqueness of Platonic Solids and the
Stabilizer-Orbit theorem
Lim Kyuson1
1
Department of Mathematics & Statistics, McMaster University, 1280 Main
St W, Hamilton, Ontario L8S 4L8, Canada, Email: limk15@mcmaster.ca
January 3, 2022
Abstract
There are 5 Platonic solids that are tetrahedron, cube, octahedron, dodecahedron, and
icosahedron. In this writing, there are four parts mainly used to describe Platonic Solids.
First, an angular defect is presented for Platonic solids to understand a deficit angle in a
planar figure. Second, Platonic Solids satisfy the Euler’s formula is presented. Third, the
Gauss-Bonnet’s formula for a discrete surface combines both concepts of Euler’s formula and
angular defect summarize the uniqueness of Platonic solids. Lastly, the Stabilizer-Orbit theorem
related to Platonic solids is presented to show how to find the order of group actions.
Introduction
Platonic Solids are 3D solids with 3 properties. Notice, the Platonic solids are the only
possible regular polyhedron that are existed. There exists no other polyhedron that satisfy
following properties (Hartshorne, R., 2013).
1. All the dihedral angles and the solid angles are congruent.
2. All the faces are composed of same regular polygons.
3. All the vertices are surrounded by the same number of faces.
There are 5 Platonic solids that are tetrahedron, cube, octahedron, dodecahedron, and
icosahedron.
Figure 1: Platonic Solids
1
In this paper, each property is carefully examined by the mathematical formulation and
group theory. Also, combinatorial understanding in Euler’s formula is examined for platonic
solids. Finally, stabilizer-orbit theorem is presented for notion of a wallpaper group to show
how 5 platonic solids satisfies for unique algebraic properties.
History and Applications
Five Platonic solids were known from the ancient Greeks philosopher Plato of his writing
Timaeus, which take into account of the formation of the universe. Then, Euclid wrote
Platonic solids in his book, the ‘Elements’, book 13th, (Mueller, I., 1981).
Often, in chemistry, molecules resemble the shape of Platonic solids. The famous greenhouse gas methane, which is caused by the decomposition of organic matters, has a shape
of a tetrahedron.
Furthermore, a table salt has a shape of a cube due to a specific arrangement by the ionic
bond. These ionic bonds let table salts melt into a water to taste in which this metabolism
comes from cubical structural construction.
A soccer ball is an example of a truncated icosahedron which is an icosahedron where all
vertices are sliced off.
Angular defect of Platonic solids
Definition. A dihedral angle is the angle of intersection of two adjacent faces while a solid angle
is the angle between two edges meeting in a vertex of a face (Sosinskiı̆, A. B., 2012).
Definition. An angular defect is a difference between 2π and the sum of angles of nearby
faces at a vertex (Sosinskiı̆, A. B., 2012).
If the 3D regular polyhedron is flattened into a 2D planar figure with respect to the vertex
chosen, then the angular defect is a deficit angle between the sum of angles from surrounding
faces and 2π. In other words, for any 2D planar figures to be assembled into a 3D polyhedron, there exists some angular defect degree left around the chosen vertex of surrounding
faces to bind each other. (Figure 2)
Angular defect P
P
Figure 2. Tetrahedron, planar figure
2
In the figure, for a vertex, P , there exists an angular defect which is a subtraction of 2π and
the sum of 3 nearby faces with an angle, π.
Proposition. An angular defect of Platonic solids is positive (Sosinskiı̆, A. B., 2012).
Proof. Let l be the number of nearby faces of a vertex P . The property 2 and 3 is
satisfied for l because of Platonic solids. Observe that any regular polyhedron in a 3D figure
has at least 3 faces that meet on a vertex.
For any regular k-gon, the angular defect is written as
2π
>0
2π–l π −
k
Now, consider l, k ≥ 3 that satisfies the equation.
Solving for l, and then k to restrict both l, k to deduce some possible regular polyhedron.
Notice, the solid angle:
2π
π
≤π−
≤ 2π
3
k
as k-gon is at least a triangle.
2
l
2
1
≤ 1 − such that, ≤ l 1 −
<2
3
k
3
k
which gives l = {3, 4, 5}.
This restricts faces meeting on any vertices. For k, any regular polyhedron must have 3 faces
that meet in a vertex,
2
2
2
2
3 l−
≤l 1−
≤2⇒1− ≤
k
k
k
3
such that k = {3, 4, 5}.
Now, consider each (l, k), (3,3) is a tetrahedron, (3,4) is a cube, (4,3) is an octahedron,
(3,5) is a dodecahedron, and (5,3) is an icosahedron. However, (4,4) is theoretically impossible because 4 squares meeting on a vertex result in 0 angular defect. Other options (4,5),
(5,4) and (5,5) are impossible due to negative angular defect.
Euler’s formula
Theorem. Let V be the number of vertices, E be the number of edges and F be the number
of faces. If a connected, planar graph is presented in a plane without any edges intersected,
then it must satisfy
V − E + F = 2 (Euler’s formula)
The Euler’s formula is applicable for Platonic solids (Ballabh, A., Trivedi, D. R., & Dastidar,
P., 2005). There exists Platonic graphs which are the planar graphs of 3D Platonic solids.
3
Definition Platonic graphs are 2-dimensional, regular graphs of Platonic solids.
Example. Consider octahedron and the planar graph.
y0
y
u
k
s
k0
o0
o
s0
u0
n0
n
Figure 3. Octahedron and planar graph
With respect to a point k and o, points s, u is stretched down to form a triangle of y, s, u
and be flattened with respect to a face y, k, o. This leads octahedron to become a planar
graph. Both have the same number of vertices and edges.
Notice, for any vertices, there exists a vertex that matches with the vertex chosen. Likewise,
the degree of any vertex is the same as the degree of the corresponding vertex in the planar
graph which is 4. By the definition, the two graphs are the same.
An edge is created by 2 vertices. If two faces meet at a vertex, then faces share an edge.
Therefore, counting the total number of nearby faces for all vertices is the same as counting
twice the total number of edges in a regular polyhedron. The property 2 and 3 from introduction is satisfied.
Lemma. Let l be the number of nearby faces on any vertex. Then,
2E = lV
For each face of a regular polyhedron, each face has the same number of edges. Since two
adjacent faces share an edge, counting the total number of edges for all faces is the same as
counting twice the total number of edges (Ballabh, A., Trivedi, D. R., & Dastidar, P., 2005).
Lemma. Let the k-gon (polygon) be a face. Then,
kF = 2E
Proof. of Euler’s formula There exists at least 6 total number of edges for a regular
polyhedron. (ie. tetrahedron)
2E
2E
and F =
V =
l
k
4
By Euler’s formula,
2 2
2 2
+ −1 ≥6
+ −1
2=E
l
k
l
k
Divide both side by 6 and rearrange.
⇔
1
2 2
2
1 1
≥ + −1⇔ ≥ +
3
l
k
3
l
k
Now, divide both sides by E:
2
2 2
1
1 1
2
= + −1⇔ < + ≤
E
l
k
2
l
k
3
1
1
1
1 1
1
For k ≥ 3, ≥ such that > − =
3
l
k
2 3
6
Therefore, 3 ≤ k < 6
1 1
1
2 1
Also, − ≤ < − ⇔ 3 ≤ l < 6
2 3
k
3 3
0<
This gives the same result from the angular defect in which possible combinations are restricted to Platonic solids. Both results show that there are no other regular polyhedron
other than Platonic solids.
Gauss Bonnet’s Formula for a Discrete Surface Without Boundary
Now, both Euler’s formula and the angular defect is combined to get the total sum of angular
defect for all vertices to be 4π (Gross, J. L., & Yellen, J., 2003).
Theorem. (Gauss-Bonnet formula for a discrete surface) For simplicity, denote d(v) as an
angle defect for a vertex v. Let X(s) be Euler’s formula for a planar graph on a discrete
surface (Ballabh, A., Trivedi, D. R., & Dastidar, P., 2005).
X
d(v) = 2πX(s)
v∈V
Proof. Let s(v) be the sum of angles for nearby faces on a vertex.
X
d(v) = 2π − s(v) and 2πX(s) = 4π,
2π − s(v) = 4π
v∈V
l
1
2π
. For all vertices, 2πV –2πV
−
Angular defect: 2π–l π −
k
2 k
2 V2l
E
l
F
l
From lemma,
= ⇒F =
⇔
= , Substitute, 2πV − 2π(E − F ) = 4π
V
2
k
V
k
Thus, the sum of an angular defect for all vertices is 4π in Platonic solids. A table chart is
provided below.
5
polyhedron
tetrahedron
F
4
E
6
V
4
k
3
l
3
d(v)
π
cube
6
12
8
4
3
2π
3
octahedron
8
12
6
3
4
π
2
dodecahedron
12
30
20
5
3
π
5
icosahedron
20
30
12
3
5
π
3
Stabilizer-Orbit theorem
This section of the essay is to extend the idea of motions and provide understandings of
group actions in Platonic solids that mainly relies on the Stabilizer-Orbit theorem. For any
transformation group, G, acting on Platonic solids, the order of |G| is easily found using
Stabilizers and Orbits.
Definition. An isometry of R3 is a function from R3 onto R3 that preserves distance.
A transformation group, G, which acts on a set, Platonic solids, the following isometry group
is composed of rotations and symmetries. Therefore, the stabilizer and the orbit makes it
easier to find isometries of Platonic solids for motions of transformation groups (Sosinskiı̆,
A. B., 2012).
Definition. Consider a transformation group g ∈ G acting on a set X, x ∈ X. The Orbit is
a set of all possible elements in X to which x can be moved by the element of G (Sosinskiı̆,
A. B., 2012).
Orb(x) = {xg | g ∈ G} ⊂ X
Definition. The Stabilizer is a set of all elements in G, which do not move x when acting on
x (Sosinskiı̆, A. B., 2012).
Stb(x) = {g ∈ G | xg = x} ⊂ G
Theorem. (Stabilizer-Orbit theorem) For a group action φ : G → Orb(x) ⇔ φ(g) = gx. The
φ is surjective because x was acted on, by all elements of G (Gallian, J. A., 2010). Notice,
the Stabilizer is a subgroup where, all g,h∈ G
φ(g) = gx = hx ⇔ φ(h) ⇔ g −1 h ∈ Stb(x).
Therefore, there is a well-defined bijection G/Stb(x)1 7→Orb(x) that is gStb(x)7→ gx. So, the
Orb(x) has the same number of elements as G/Stb(x) by Lagnage’s theorem (Gallian, J. A.,
2010).
|G|
⇔ |G| = |Orb(x)||Stb(x)|
|Orb(x)| =
|Stb(x)|
1
This is a coset
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Notice, the length of the orbit of x, times the order of its stabilizer is the order of the group
due to identical vertices (Sosinskiı̆, A. B., 2012).
Theorem. (Lagrange theorem) For any finite group G, the order of every subgroup H of G
divide the order of G (Gallian, J. A., 2010).
Example. Consider a cube that has 48 isometries (Sosinskiı̆, A. B., 2012). A cube has 8
vertices that are elements of Orb(x). (Fig.4.1)
s
s
u
k
o
n
o
y
l
u
k
i
n
y
l
i
Figure 4. Cube, Rotation 23 π, Reflection
Now, choose any vertices but for simplicity let the chosen vertex be E.
From the theorem, |G| = |8||Stb(k)|
If the vertex k is fixed, one of the neighbourhood vertices s, o, i would be moved. In other
words, any element of G that fixes k must send s to either o, i or s.
|Stb(k)| = |Stb(Stb(k), s)||Orb(Orb(k), s)|
Now, consider the rotation around the diagonal axis through k and y by 32 π which permutes
s, o, i and l, u, n, and fixes k and y. (Fig. 4.2)
Thus, |Orb(Orb(k), s)| = 3.
Apply the theorem again for Stb(Stb(k), s).2 Then, any element of G that fixes k and s must
send u to either u or n. Reflection of the cube through the plane through k, s, y and l sends
u to n.(Fig. 4.3)
Thus, |Orb(Orb(Orb(k), s), u)| = 2.
Also, the |Stb(x)| 3 fixing k, s and u must also fix all other vertices, since they are determined
by neighbourhood vertices. Thus, it has an order of 1.
So, |G|4 = 8 · 3 · 2 · 1 = 48
2
Stabilizer of vertices k, s fixed is |Stb(Stb(k), s)| = |Stb(Stb(Stb(k), s, u))| · |Orb(Orb(Orb(k), s, u))|
Then, Stabilizer fixes vertices k, s, u, which is |Stb(Stb(Stb(k), s, u))|
4
The decomposition is |G| = |Orb(k)| · |Orb(Orb(k), s)| · |Orb(Orb(Orb(k), s), u)| · |Stb(Stb(Stb(k), s, u))|
3
7
A structural understanding becomes easier with decomposition of formula. By the example,
where points move and elements of the transformation group becomes predictable. The
Stabilizer-Orbit theorem extends knowledge of transformation groups and motions what to
do and where points move for Platonic solids (Gallian, J. A., 2010).
Conclusion
A concept of the angular defect and Euler’s formula provides uniqueness of Platonic solids
from given the properties. Then, both concepts are combined for Platonic solids to follow
Gauss-Bonnet’s formula. This results in a sum of total degree for all vertices of angular
defect be equal to 4π, only for Platonic solids.
Lastly, the Orbit-Stabilizer theorem gives an idea of motions related to group actions of
Platonic solids. After all, any orders of transformation groups acting on Platonic solids are
able to be deduced.
References
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